Equivariant K-theory

For the topological equivariant K-theory, see topological K-theory.

In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category $\operatorname{Coh}^G(X)$ of equivariant coherent sheaves on an algebraic scheme X with action of a linear algebraic group G, via Quillen's Q-construction; thus, by definition,

$K_i^G(X) = \pi_i(B^+ \operatorname{Coh}^G(X)).$

In particular, $K_0^G(C)$ is the Grothendieck group of $\operatorname{Coh}^G(X)$. The theory was developed by R. W. Thomason in 1980s.^[1] Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

Equivalently, $K_i^G(X)$ may be defined as the $K_{i}$ of the category of coherent sheaves on the quotient stack $[X/G]$. (Hence, the equivariant K-theory is a specific case of the K-theory of a stack.)

A version of the Lefschetz fixed point theorem holds in the setting of equivariant (algebraic) K-theory.^[2]

Fundamental theorems

Let X be an equivariant algebraic scheme.

Localization theorem — Given a closed immersion $Z \hookrightarrow X$ of equivariant algebraic schemes and an open immersion $Z - U \hookrightarrow X$, there is a long exact sequence of groups

$\dots \to K^G_i(Z) \to K^G_i(X) \to K^G_i(U) \to K^G_{i-1}(Z) \to \dots$

References

1. Charles A. Weibel, Robert W. Thomason (1952–1995).
2. BFQ 1979

• N. Chris and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
• Baum, P., Fulton, W., Quart, G.: Lefschetz Riemann Roch for singular varieties. Acta. Math. 143, 193-211 (1979)
• Thomason, R.W.:Algebraic K-theory of group scheme actions. In: Browder, W. (ed.) Algebraic topology and algebraic K-theory. (Ann. Math. Stud., vol. 113, pp. 539 563) Princeton: Princeton University Press 1987
• Thomason, R.W.: Lefschetz-Riemann-Roch theorem and coherent trace formula. Invent. Math. 85, 515-543 (1986)
• Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. In: Cartier, P., Illusie, L., Katz, N.M., Laumon, G., Manin, Y., Ribet, K.A. (eds.) The Grothendieck Festschrift, vol. III. (Prog. Math. vol. 88, pp. 247 435) Boston Basel Berlin: Birkhfiuser 1990
• Thomason, R.W., Une formule de Lefschetz en K-théorie équivariante algébrique, Duke Math. J. 68 (1992), 447-462.

Further reading

• Dan Edidin, Riemann–Roch for Deligne–Mumford stacks, 2012